[Paper Review] New proofs of the duplication and multiplication formulae for the gamma and the Barnes double gamma functions

Donal F. Connon|ArXiv.org|Mar 26, 2009

Advanced Mathematical Identities11 references9 citations

TL;DR

This paper presents new proofs of the duplication and multiplication formulas for the gamma and Barnes double gamma functions using the Hurwitz zeta function. It provides concise derivations of Gauss's multiplication theorem for the gamma function and its double gamma counterpart, highlighting connections to Stieltjes constants.

ABSTRACT

New proofs of the duplication formulae for the gamma and the Barnes double gamma functions are derived using the Hurwitz zeta function. Concise derivations of Gauss's multiplication theorem for the gamma function and a corresponding one for the double gamma function are also reported. This paper also refers to some connections with the Stieltjes constants.

Motivation & Objective

• To provide novel, concise derivations of the duplication and multiplication formulae for the classical gamma function.
• To extend these derivations to the Barnes double gamma function using the same analytical framework.
• To establish connections between the derived formulae and the Stieltjes constants through the Hurwitz zeta function.
• To offer a unified approach based on zeta function techniques for fundamental identities in special functions.
• To improve understanding of the structural properties of multiple gamma functions via zeta regularization.

Proposed method

• Utilizes the Hurwitz zeta function as the central analytical tool to derive functional identities.
• Applies integral representations and series expansions related to the Hurwitz zeta function to manipulate gamma function products.
• Employs the reflection and multiplication properties of the Hurwitz zeta function to derive the duplication and multiplication theorems.
• Applies zeta regularization techniques to handle divergent series and infinite products inherent in multiple gamma functions.
• Establishes links between the derived formulae and the Stieltjes constants via Laurent series expansions of the zeta function.
• Uses analytic continuation and functional equations of the Hurwitz zeta function to validate the results across the complex plane.

Experimental results

Research questions

• RQ1 How can the duplication formula for the gamma function be re-derived using the Hurwitz zeta function?
• RQ2 What is the analogous duplication formula for the Barnes double gamma function, and how can it be proven via zeta function methods?
• RQ3 Can Gauss’s multiplication theorem for the gamma function be derived concisely using zeta function techniques?
• RQ4 What connections exist between the derived multiplication formulae and the Stieltjes constants?
• RQ5 To what extent can the Hurwitz zeta function unify the derivation of multiplication and duplication identities for multiple gamma functions?

Key findings

• New, concise proofs of the duplication formula for the classical gamma function are derived using the Hurwitz zeta function.
• A corresponding duplication formula for the Barnes double gamma function is established through the same zeta-based method.
• Gauss’s multiplication theorem for the gamma function is re-derived in a streamlined manner using zeta function identities.
• An analogous multiplication formula for the Barnes double gamma function is derived using the same framework.
• The paper reveals explicit connections between the derived functional equations and the Stieltjes constants via Laurent series of the Hurwitz zeta function.
• The results demonstrate the utility of the Hurwitz zeta function as a unifying tool for deriving identities in multiple gamma function theory.